Logistic growth is one of 1 models formula in the differential equations section of this library, and it is used at ap · university level.
Why logistic growth works
The factor (1 − P/K) acts as a brake. While the population is small that factor is close to 1, so growth is nearly exponential, and as the population approaches K the factor slides toward 0 and the growth stalls. The solution formula is what you get after separating variables and clearing the resulting logarithms.
What each symbol means
$K$ is carrying capacity, $r$ intrinsic rate, and $A=(K-P_0)/P_0$.
Logistic growth: when it holds
$K,r,P_0>0$ for the usual population model.
When it stops applying
The model assumes a fixed carrying capacity and instant response to crowding. Real populations with breeding delays can overshoot K and oscillate around it, and a harvest or a changing habitat moves K, so this fixed-K curve stops tracking the data.
Logistic growth: a worked example
At $P=K/2$, growth rate is $rK/4$, its maximum.
The mistake to avoid
What people do: Students substitute the starting population directly for A in the solution.
Why it goes wrong: A is not the initial population; it is the ratio (K − P₀)/P₀. Using P₀ itself puts the curve at the wrong height at t = 0.
Do this instead: Compute A from the starting values first. With K = 1000 and P₀ = 100 you get A = 9, and checking t = 0 gives 1000/(1 + 9) = 100, which confirms the setup.
Logistic growth: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $K$ is carrying capacity, $r$ intrinsic rate, and $A=(K-P_0)/P_0$.
- Check the conditions before substituting. $K,r,P_0>0$ for the usual population model.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most differential equations slips.
Where this formula fits
- Subject
- Differential Equations formulas — 8 entries in this library
- Topic
- Models
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where logistic growth comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Exponentials and Logarithms — the lesson behind this formula: move between growth models and inverse logarithmic form.
- Calculus Calculator — check your substitution and the value it produces.
- Study differential equations — the subject guide that explains the ideas these formulas compress.
- Differential Equations Practice — questions that make you retrieve the formula instead of recognising it.
- All 8 differential equations formulas — the full grouped reference, or the complete formula library.
Questions about logistic growth
When is the population growing fastest?
Right at half the carrying capacity. Substituting P = K/2 gives a growth rate of rK/4, so with r = 0.5 and K = 1000 the peak rate is 125 per unit time, and that point is where the S-curve changes its bend.
What happens if the population starts above K?
It falls toward K instead of rising. Then A comes out negative, the formula still applies, and the curve approaches the same ceiling from above.
How is this different from plain exponential growth?
Exponential growth has no ceiling and eventually predicts absurd numbers. Logistic growth matches it early on, then bends over and levels off, which is why it fits real populations far better over long spans.
Does the population ever actually reach K?
Not in finite time. The exponential term shrinks toward zero but never gets there, so the curve approaches the carrying capacity as a limit and only meets it in the long run.